Understanding Charged Particle Deflection by Magnetic Fields

When a charged particle moves through a magnetic field, it experiences a force. This force is known as the Lorentz force. The magnitude and direction of this force depend on the charge of the particle, its velocity, and the magnetic field.

The formula for the Lorentz force (\(\vec{F}\)) on a charged particle with charge \(q\) and velocity \(\vec{v}\) in a magnetic field \(\vec{B}\) is given by:

\(\vec{F} = q(\vec{v} \times \vec{B})\)

In this formula:

• \(\vec{F}\) is the magnetic force vector.
• \(q\) is the electric charge of the particle.
• \(\vec{v}\) is the velocity vector of the particle.
• \(\vec{B}\) is the magnetic field vector.
• \(\times\) denotes the cross product of the two vectors.

The direction of the force is perpendicular to both the velocity vector (\(\vec{v}\)) and the magnetic field vector (\(\vec{B}\)). For a positive charge (\(q > 0\)), the direction of the force \(\vec{F}\) is the same as the direction of the cross product \(\vec{v} \times \vec{B}\). For a negative charge (\(q < 0\)), the direction of the force \(\vec{F}\) is opposite to the direction of the cross product \(\vec{v} \times \vec{B}\).

Analyzing the Charged Particle's Motion and Force

In the given problem, we have a positively charged particle. Let's note down the given information:

• Charge of the particle: Positive (\(q > 0\))
• Direction of velocity (\(\vec{v}\)): Towards East
• Direction of deflection force (\(\vec{F}\)): Towards North

Since the particle is positively charged, the direction of the force \(\vec{F}\) is the same as the direction of the cross product \(\vec{v} \times \vec{B}\). Thus, we have:

Direction of \(\vec{v} \times \vec{B}\) is North.

Determining the Magnetic Field Direction using the Right-Hand Rule

We need to find the direction of the magnetic field \(\vec{B}\) such that when the velocity \(\vec{v}\) (East) is crossed with \(\vec{B}\), the resulting direction is North. We can use the right-hand rule for cross products to determine this.

The right-hand rule states: If you point the fingers of your right hand in the direction of the first vector (\(\vec{v}\)) and curl them towards the direction of the second vector (\(\vec{B}\)), your thumb will point in the direction of the resulting vector (\(\vec{v} \times \vec{B}\)).

In our case:

• First vector (\(\vec{v}\)) direction: East
• Resulting vector (\(\vec{v} \times \vec{B}\)) direction: North
• We need to find the direction of the second vector (\(\vec{B}\)).

Let's apply the right-hand rule by pointing the fingers of our right hand towards East and our thumb towards North. Now, observe the direction in which your fingers would naturally curl to reach your thumb's direction. If your fingers point East and thumb points North, your fingers must curl downwards. This indicates that the direction of the magnetic field \(\vec{B}\) must be Downward.

Evaluating the Options for Magnetic Field Direction

Let's verify this result by checking the cross product for each given option for the magnetic field direction:

As shown in the table, only when the magnetic field is directed Downward does the cross product \(\vec{v} \times \vec{B}\) point towards North, which matches the observed deflection force direction for a positively charged particle.

Conclusion on Magnetic Field Direction

Based on the Lorentz force formula and the right-hand rule, for a positively charged particle moving East to be deflected North, the magnetic field must be directed Downward.

Revision Table: Magnetic Force on Charged Particle

Additional Information: Charged Particle Motion in Magnetic Fields

When a charged particle enters a uniform magnetic field, the force it experiences is always perpendicular to its velocity. This perpendicular force does not change the speed of the particle, but it changes its direction, causing the particle to move in a curved path.

• If the velocity is perpendicular to the magnetic field, the particle moves in a circle.
• If the velocity is parallel or anti-parallel to the magnetic field, the force is zero, and the particle moves in a straight line.
• If the velocity has components both parallel and perpendicular to the magnetic field, the particle moves in a helical path.

The radius of the circular path (when \(\vec{v} \perp \vec{B}\)) is given by \(r = \frac{mv}{|q|B}\), where \(m\) is the mass of the particle, \(v\) is its speed, \(|q|\) is the magnitude of its charge, and \(B\) is the magnetic field strength.